It’s Nov. 11, “Pepero Day” in South Korea, when the marketing folks at Lotte Confectionery Co. would like you to show your affection for friends and loved-ones by exchanging gifts of their chocolate-coated wand-like pretzel snack.

The made-to-order holiday has been a resounding success for Lotte over the years, but there’s a complication this year: a rival product with the same promotional trick.

Legend has it that Pepero Day began in 1994 when middle-school girls in the southern city of Busan decided to give each other Pepero on 11/11 with wishes to become skinny—like the snack stick or the four 1’s that make up the date.

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And by legend, we mean it’s a story that has been circulated in the local media and cited by Lotte, but we were unable to find anyone who could offer proof. (We are also skeptical that anyone believes eating chocolate makes you thin.)

Exactly five years after that fabled fall event, a short hop away in Japan, Ezaki Glico Co. launched “Pocky Day.” Pocky, for the uninitiated, looks pretty much exactly like Pepero: a thin, straight pretzel stick with a range of chocolate coatings.

Steve Miller, a blogger in South Korea, has a useful guide to the subtle differences in the treats.

Pocky Day began in Japan in 1999, or year 11 of the reign of the current Japanese emperor, meaning the full date is written 11/11/11. Kyoko Yamamoto, a Glico public relations official, says that in addition to having the stick-shaped date, November had been a lean month for promotional events so the company decided to adopt it.

Alastair Gale/The Wall Street Journal

Display stands for Pocky (left) and Pepero (right) in a supermarket in Seoul.

Despite the matching sales gimmick, neither company had much to worry about because Pocky wasn’t sold in Korea and Pepero wasn’t sold in Japan. That changed this summer, when Glico set up a joint Pocky venture in South Korea.

Ms. Yamamoto says that Glico needed new markets as Japan’s society ages and its sweet tooth dulls. To give its sales in Korea a kick-start, Glico signed up K-Pop group Girl’s Day for a promotional video (slogan: “Happy Together, Pocky Together”) and has held sponsored ‘fan days’ where fans can meet the group.

Pocky displays can now be found next to those for Pepero in Korean supermarkets and department stores. And while it would be an exaggeration to say a turf war has broken out, Lotte is definitely nonplused.

“Pocky Day is a copy of Pepero Day,” said Jeong Sung-won, a Lotte official.

The company has good reason to defend its Nov. 11 monopoly: About 50% of Pepero’s annual sales of some 87 billion won ($81.6 million) come between September and November, Mr. Jeong said.

Glico’s Ms. Yamamoto says it’s too soon to assess Pocky sales in Korea, but the company’s local joint venture is aiming for revenue of 50 billion won in 2018.

The twist in the tale is that Glico first came up with the idea for a chocolate-dipped pretzel stick in 1966, and Lotte didn’t introduce the Pepero until 1983. At the time, Glico considered taking action against what it considered a copycat snack, says Ms. Yamamoto—but found it would be difficult, because Pocky wasn’t sold in Korea.

Lotte, naturally, denies it was inspired by Glico.

“There is no relation to Pocky at all,” said Lotte’s Mr. Jeong.

Glico emphasizes that Pocky sold in South Korea is made locally and rejects suggestions that it might inflame nationalist tensions at a time when Tokyo and Seoul are at loggerheads over territorial and historical issues.

“We won’t step into any political issues,” said Ms. Yamamoto.

Nonetheless, competition between Pepero and Pocky may heat up in coming years as the companies look to expand their reach in more markets. Both are entering more South East Asian countries, for example.

Lotte says it will be bringing its Pepero mascots to the Bugis+ shopping center in Singapore today, so if you’re intrigued by cuddly characters with massively elongated heads, you know where to find them.